On the existence of almost greedy bases in Banach spaces

S. J. Dilworth ¹   ; N. J. Kalton ²   ; Denka Kutzarova ³

¹ Department of Mathematics University of South Carolina Columbia, SC 29208, U.S.A.
² Department of Mathematics University of Missouri Columbia, MO 65211, U.S.A.
³ Institute of Mathematics Bulgarian Academy of Sciences Sofia, Bulgaria and Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL 61801, U.S.A.

Studia Mathematica, Tome 159 (2003) no. 1, pp. 67-101

Résumé

We consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the Thresholding Greedy Algorithm (TGA). In particular, we continue the study of almost greedy bases begun in [3]. We show that almost greedy bases are essentially optimal for $n$-term approximation when the TGA is modified to include a Chebyshev approximation. We prove that if a Banach space $X$ has a basis and contains a complemented subspace with a symmetric basis and finite cotype then $X$ has an almost greedy basis. We show that $c_0$ is the only ${\mathcal L}_\infty $ space to have a quasi-greedy basis. The Banach spaces which contain almost greedy basic sequences are characterized.

Détail

DOI : 10.4064/sm159-1-4

Keywords: consider several greedy conditions bases banach spaces arise naturally study thresholding greedy algorithm tga particular continue study almost greedy bases begun almost greedy bases essentially optimal n term approximation tga modified include chebyshev approximation prove banach space has basis contains complemented subspace symmetric basis finite cotype has almost greedy basis only mathcal infty space have quasi greedy basis banach spaces which contain almost greedy basic sequences characterized